Perturbative world-volume dynamics of the bosonic membrane and string

TL;DR

This paper investigates the perturbative world-volume dynamics of the bosonic membrane, explores conditions for critical dimensions, and uses string reductions to infer the membrane's critical dimension as 27.

Contribution

It constructs a suitable action for perturbative analysis of the bosonic membrane and deduces its critical dimension through string model reductions.

Findings

Critical dimension for the membrane is inferred as 27.

String models obtained from membrane reduction confirm critical dimension as 26 for strings.

No explicit background satisfying the conditions for perturbative expansion was found.

Abstract

We study the world-volume theory of a bosonic membrane perturbatively and discuss if one can obtain any conditions on the number of space-time dimensions from the consistency of the theory. We construct an action which is suitable for such a study. In order to study the theory perturbatively we should specify a classical background around which perturbative expansion is defined. We will discuss the conditions which such a background should satisfy to deduce the critical dimension. Unfortunately we do not know any background satisfying such conditions. In order to get indirect evidences for the critical dimension of the membrane, we next consider two string models obtained via double dimensional reduction of the membrane. The first one reduces to the Polyakov string theory in the conformal gauge. The second one is described by the Schild action. We show that the critical dimension is 26 for these string theories, which implies that the critical dimension is 27 for the membrane theory.